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In algebraic number theory the n-th power residue symbol (for an integer n > 2) is a generalization of the (quadratic) Legendre symbol to n-th powers. These symbols are used in the statement and proof of cubic, quartic, Eisenstein, and related higher^[1] reciprocity laws.^[2]

Background and notation

Let k be an algebraic number field with ring of integers ${\mathcal {O}}_{k}$ that contains a primitive n-th root of unity $\zeta _{n}\in {\mathcal {O}}_{k}.$

Let ${\mathfrak {p}}\subset {\mathcal {O}}_{k}$ be a prime ideal and assume that n and ${\mathfrak {p}}$ are coprime (i.e. $n\not \in {\mathfrak {p}}$ .)

The norm of ${\mathfrak {p}}$ is defined as the cardinality of the residue class ring ${\mathcal {O}}_{k}/{\mathfrak {p}}\;:\;\;\;\mathrm {N} {\mathfrak {p}}=|{\mathcal {O}}_{k}/{\mathfrak {p}}|.$ (since ${\mathfrak {p}}$ is prime this is a finite field)

There is an analogue of Fermat's theorem in ${\mathcal {O}}_{k}:$ If $\alpha \in {\mathcal {O}}_{k},\;\;\;\alpha \not \in {\mathfrak {p}},$ then

\alpha ^{\mathrm {N} {\mathfrak {p}}-1}\equiv 1{\pmod {\mathfrak {p}}}.

And finally, $\mathrm {N} {\mathfrak {p}}\equiv 1{\pmod {n}}.$ These facts imply that

\alpha ^{\frac {\mathrm {N} {\mathfrak {p}}-1}{n}}\equiv \zeta _{n}^{s}{\pmod {\mathfrak {p}}}

is well-defined and congruent to a unique n-th root of unity ζ_n^s.

Definition

This root of unity is called the n-th power residue symbol for ${\mathcal {O}}_{k},$ and is denoted by

\left({\frac {\alpha }{\mathfrak {p}}}\right)_{n}=\zeta _{n}^{s}\equiv \alpha ^{\frac {\mathrm {N} {\mathfrak {p}}-1}{n}}{\pmod {\mathfrak {p}}}.

Properties

The n-th power symbol has properties completely analogous to those of the classical (quadratic) Legendre symbol:

\left({\frac {\alpha }{\mathfrak {p}}}\right)_{n}={\begin{cases}0&{\mbox{ if }}\alpha \in {\mathfrak {p}}\\1&{\mbox{ if }}\alpha \not \in {\mathfrak {p}}{\mbox{ and there is an }}\eta \in {\mathcal {O}}_{k}{\mbox{ such that }}\alpha \equiv \eta ^{n}{\pmod {\mathfrak {p}}}\\\zeta {\mbox{ where }}\zeta ^{n}=1{\mbox{ and }}\zeta \neq 1&{\mbox{ if }}\alpha \not \in {\mathfrak {p}}{\mbox{ and there is no such }}\eta \end{cases}}

In all cases (zero and nonzero)

\left({\frac {\alpha }{\mathfrak {p}}}\right)_{n}\equiv \alpha ^{\frac {\mathrm {N} {\mathfrak {p}}-1}{n}}{\pmod {\mathfrak {p}}}.

\left({\frac {\alpha }{\mathfrak {p}}}\right)_{n}\left({\frac {\beta }{\mathfrak {p}}}\right)_{n}=\left({\frac {\alpha \beta }{\mathfrak {p}}}\right)_{n}

{\mbox{if }}\alpha \equiv \beta {\pmod {\mathfrak {p}}}{\mbox{ then }}\left({\frac {\alpha }{\mathfrak {p}}}\right)_{n}=\left({\frac {\beta }{\mathfrak {p}}}\right)_{n}

Relation to the Hilbert symbol

The n-th power residue symbol is related to the Hilbert symbol $(\cdot ,\cdot )_{\mathfrak {p}}$ for the prime ${\mathfrak {p}}$ by

\left({\frac {\alpha }{\mathfrak {p}}}\right)_{n}=\left({\pi ,\alpha }\right)_{\mathfrak {p}}

in the case ${\mathfrak {p}}$ coprime to n, where $\pi$ is any uniformising element for the local field $K_{\mathfrak {p}}$ .^[3]

Generalizations

The n-th power symbol may be extended to take non-prime ideals or non-zero elements as its "denominator", in the same way that the Jacobi symbol extends the Legendre symbol.

Any ideal ${\mathfrak {a}}\subset {\mathcal {O}}_{k}$ is the product of prime ideals, and in one way only:

{\mathfrak {a}}={\mathfrak {p}}_{1}{\mathfrak {p}}_{2}\dots {\mathfrak {p}}_{g}.

The n-th power symbol is extended multiplicatively:

{\bigg (}{\frac {\alpha }{\mathfrak {a}}}{\bigg )}_{n}=\left({\frac {\alpha }{{\mathfrak {p}}_{1}}}\right)_{n}\left({\frac {\alpha }{{\mathfrak {p}}_{2}}}\right)_{n}\dots \left({\frac {\alpha }{{\mathfrak {p}}_{g}}}\right)_{n}.

If $\beta \in {\mathcal {O}}_{k}$ is not zero the symbol $\left({\frac {\alpha }{\beta }}\right)_{n}$ is defined as

\left({\frac {\alpha }{\beta }}\right)_{n}=\left({\frac {\alpha }{(\beta )}}\right)_{n},

where

(\beta )

is the principal ideal generated by

\beta .

The properties of this symbol are analogous to those of the quadratic Jacobi symbol:

{\mbox{If }}\alpha \equiv \beta {\pmod {\mathfrak {a}}}{\mbox{ then }}{\bigg (}{\frac {\alpha }{\mathfrak {a}}}{\bigg )}_{n}=\left({\frac {\beta }{\mathfrak {a}}}\right)_{n}.

{\bigg (}{\frac {\alpha }{\mathfrak {a}}}{\bigg )}_{n}\left({\frac {\beta }{\mathfrak {a}}}\right)_{n}=\left({\frac {\alpha \beta }{\mathfrak {a}}}\right)_{n}.

\left({\frac {\alpha }{\mathfrak {a}}}\right)_{n}\left({\frac {\alpha }{\mathfrak {b}}}\right)_{n}=\left({\frac {\alpha }{\mathfrak {ab}}}\right)_{n}.

{\mbox{If }}\alpha \equiv \eta ^{n}{\pmod {\mathfrak {a}}}{\mbox{ then }}\left({\frac {\alpha }{\mathfrak {a}}}\right)_{n}=1.

{\mbox{If }}\left({\frac {\alpha }{\mathfrak {a}}}\right)_{n}\neq 1{\mbox{ then }}\alpha {\mbox{ is not an }}n{\mbox{-th power}}{\pmod {\mathfrak {a}}}.

{\mbox{If }}\left({\frac {\alpha }{\mathfrak {a}}}\right)_{n}=1{\mbox{ then }}\alpha {\mbox{ may or may not be an }}n{\mbox{-th power}}{\pmod {\mathfrak {a}}}.

Power reciprocity law

The power reciprocity law, the analogue of the law of quadratic reciprocity, may be formulated in terms of the Hilbert symbols as^[4]

\left({\frac {\alpha }{\beta }}\right)_{n}\left({\frac {\beta }{\alpha }}\right)_{n}=\prod _{{\mathfrak {p}}|n\infty }(\alpha ,\beta )_{\mathfrak {p}}\ .

Notes

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References

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Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

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To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010

Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010

↑ Quadratic reciprocity deals with squares; higher refers to cubes, fourth, and higher powers.
↑ All the facts in this article are in Lemmermeyer Ch. 4.1 and Ireland & Rosen Ch. 14.2
↑ Neukirch (1999) p. 336
↑ Neukirch (1999) p. 415

[1] Quadratic reciprocity deals with squares; higher refers to cubes, fourth, and higher powers.

[2] All the facts in this article are in Lemmermeyer Ch. 4.1 and Ireland & Rosen Ch. 14.2

[Neu336-3] Neukirch (1999) p. 336

[Neu415-4] Neukirch (1999) p. 415

[1]

[2]

[3]

[4]

Biregular graph: Difference between revisions

Latest revision as of 06:06, 13 July 2013

Contents

Background and notation

Definition

Properties

Relation to the Hilbert symbol

Generalizations

Power reciprocity law

See also

Notes

References

Navigation menu

@@ Line 1: / Line 1: @@
+In [[algebraic number theory]] the '''''n''-th power residue symbol''' (for an integer ''n'' > 2) is a generalization of the (quadratic) [[Legendre symbol]] to ''n''-th powers. These symbols are used in the statement and proof of [[cubic reciprocity|cubic]], [[quartic reciprocity|quartic]], [[Eisenstein reciprocity|Eisenstein]], and related higher<ref>[[Quadratic reciprocity]] deals with squares; higher refers to cubes, fourth, and higher powers.</ref> [[reciprocity law]]s.<ref>All the facts in this article are in Lemmermeyer Ch. 4.1 and Ireland & Rosen Ch. 14.2</ref>
+==Background and notation==
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+Let ''k'' be an [[algebraic number field]] with [[ring of integers]]  &nbsp; <math>\mathcal{O}_k</math>&nbsp; that contains a [[root of unity|primitive ''n''-th root of unity]]  &nbsp; <math>\zeta_n\in\mathcal{O}_k.</math>
+Let  &nbsp; <math>\mathfrak{p} \subset \mathcal{O}_k </math>  &nbsp; be a [[prime ideal]] and assume that ''n'' and <math>\mathfrak{p}</math> are [[coprime]] (i.e. <math>n \not \in \mathfrak{p}</math>.)
+The [[Norm of an ideal|norm]] of &nbsp;<math>\mathfrak{p} </math>&nbsp; is defined as the cardinality of the residue class ring  &nbsp; <math> \mathcal{O}_k / \mathfrak{p}\;:\;\;\; \mathrm{N} \mathfrak{p} = |\mathcal{O}_k / \mathfrak{p}|.</math> &nbsp; (since <math>\mathfrak{p} </math>  is prime this is a [[finite field]])
+There is an analogue of Fermat's theorem in &nbsp; <math>\mathcal{O}_k:</math>&nbsp; If &nbsp;  <math>\alpha \in \mathcal{O}_k,\;\;\; \alpha\not\in \mathfrak{p},</math> &nbsp; then
+:<math>\alpha^{\mathrm{N} \mathfrak{p} -1}\equiv 1 \pmod{\mathfrak{p} }.
+</math>
+And finally, &nbsp;  <math>\mathrm{N} \mathfrak{p} \equiv 1 \pmod{n}.</math> &nbsp;
+These facts imply that
+:<math>\alpha^{\frac{\mathrm{N} \mathfrak{p} -1}{n}}\equiv \zeta_n^s\pmod{\mathfrak{p} }
+</math> &nbsp; is well-defined and congruent to a unique ''n''-th root of unity &zeta;<sub>''n''</sub><sup>''s''</sup>.
+==Definition==
+This root of unity is called  the '''''n''-th power residue symbol for  &nbsp; <math>\mathcal{O}_k,</math>''' &nbsp;  and is denoted by
+:<math>
+\left(\frac{\alpha}{\mathfrak{p} }\right)_n= \zeta_n^s \equiv \alpha^{\frac{\mathrm{N} \mathfrak{p} -1}{n}}\pmod{\mathfrak{p}}.
+</math>
+==Properties==
+The ''n''-th power symbol has properties completely analogous to those of the classical (quadratic) [[Legendre symbol]]:
+:<math>
+\left(\frac{\alpha}{\mathfrak{p} }\right)_n
+=
+\begin{cases}
+&\mbox{ if } \alpha\in\mathfrak{p}\\
+&\mbox{ if }\alpha\not\in\mathfrak{p}\mbox{ and there is an } \eta \in\mathcal{O}_k\mbox{ such that }\alpha\equiv\eta^n\pmod{\mathfrak{p}}\\
+\zeta \mbox{ where }\zeta^n=1\mbox{ and }\zeta \neq 1&\mbox{ if }\alpha\not\in\mathfrak{p}\mbox{ and there is no such }\eta
+\end{cases}
+</math>
+In all cases (zero and nonzero)
+:<math>
+\left(\frac{\alpha}{\mathfrak{p} }\right)_n \equiv \alpha^{\frac{\mathrm{N} \mathfrak{p} -1}{n}}\pmod{\mathfrak{p}}.
+</math>
+:<math>
+\left(\frac{\alpha}{\mathfrak{p} }\right)_n
+\left(\frac{\beta}{\mathfrak{p} }\right)_n
+=
+\left(\frac{\alpha\beta}{\mathfrak{p} }\right)_n
+</math>
+:<math>
+\mbox{if }\alpha \equiv\beta\pmod{\mathfrak{p}}
+\mbox{ then }
+\left(\frac{\alpha}{\mathfrak{p} }\right)_n
+=
+\left(\frac{\beta}{\mathfrak{p} }\right)_n
+</math>
+==Relation to the Hilbert symbol==
+The ''n''-th power residue symbol is related to the [[Hilbert symbol]] <math>(\cdot,\cdot)_{\mathfrak{p}}</math> for the prime <math>\mathfrak{p}</math> by
+:<math>\left(\frac{\alpha}{\mathfrak{p} }\right)_n = \left({\pi, \alpha}\right)_{\mathfrak{p}} </math>
+in the case <math>\mathfrak{p}</math> coprime to ''n'', where <math>\pi</math> is any [[uniformising element]] for the [[local field]] <math>K_{\mathfrak{p}}</math>.<ref name=Neu336>Neukirch (1999) p. 336</ref>
+==Generalizations==
+The ''n''-th power symbol may be extended to take non-prime ideals or non-zero elements as its "denominator", in the same way that the [[Jacobi symbol]] extends the Legendre symbol.
+Any ideal <math>\mathfrak{a}\subset\mathcal{O}_k</math> is the product of prime ideals, and in one way only:
+:<math>\mathfrak{a} = \mathfrak{p}_1 \mathfrak{p}_2 \dots\mathfrak{p}_g.
+</math>
+The ''n''-th power symbol is extended multiplicatively:
+:<math>
+\bigg(\frac{\alpha}{\mathfrak{a} }\bigg)_n
+=
+\left(\frac{\alpha}{\mathfrak{p}_1 }\right)_n
+\left(\frac{\alpha}{\mathfrak{p}_2 }\right)_n
+\dots
+\left(\frac{\alpha}{\mathfrak{p}_g }\right)_n.
+</math>
+If <math>\beta\in\mathcal{O}_k</math> is not zero the symbol <math>\left(\frac{\alpha}{\beta}\right)_n</math> is defined as
+:<math>
+\left(\frac{\alpha}{\beta}\right)_n = \left(\frac{\alpha}{(\beta) }\right)_n,
+</math> where <math>(\beta)</math> is the principal ideal generated by <math>\beta.</math>
+The properties of this symbol are analogous to those of the quadratic Jacobi symbol:
+:<math>
+\mbox{If }\alpha\equiv\beta\pmod{\mathfrak{a}}
+\mbox{ then }
+\bigg(\frac{\alpha}{\mathfrak{a} }\bigg)_n =
+\left(\frac{\beta}{\mathfrak{a} }\right)_n.
+</math>
+:<math>
+\bigg(\frac{\alpha}{\mathfrak{a} }\bigg)_n
+\left(\frac{\beta}{\mathfrak{a} }\right)_n
+=
+\left(\frac{\alpha\beta}{\mathfrak{a} }\right)_n.
+</math>
+:<math>
+\left(\frac{\alpha}{\mathfrak{a} }\right)_n
+\left(\frac{\alpha}{\mathfrak{b} }\right)_n
+=
+\left(\frac{\alpha}{\mathfrak{ab} }\right)_n.
+</math>
+:<math>
+\mbox{If } \alpha\equiv\eta^n\pmod{\mathfrak{a}}\mbox{ then }
+\left(\frac{\alpha}{\mathfrak{a} }\right)_n =1.
+</math>
+:<math>
+\mbox{If } \left(\frac{\alpha}{\mathfrak{a} }\right)_n \neq1
+\mbox{ then }\alpha \mbox{ is not an }n\mbox{-th power}\pmod{\mathfrak{a}}.
+</math>
+:<math>
+\mbox{If } \left(\frac{\alpha}{\mathfrak{a} }\right)_n =1
+\mbox{ then }\alpha \mbox{ may or may not be an }n\mbox{-th power}\pmod{\mathfrak{a}}.
+</math>
+==Power reciprocity law==
+The ''power reciprocity law'', the analogue of the [[law of quadratic reciprocity]], may be formulated in terms of the [[Hilbert symbol]]s as<ref name=Neu415>Neukirch (1999) p. 415</ref>
+:<math>\left({\frac{\alpha}{\beta}}\right)_n \left({\frac{\beta}{\alpha}}\right)_n = \prod_{\mathfrak{p} | n\infty} (\alpha,\beta)_{\mathfrak{p}} \ . </math>
+==See also==
+*[[Artin symbol]]
+*[[Gauss's lemma (number theory)|Gauss's lemma]]
+==Notes==
+{{reflist}}
+==References==
+* {{citation | last=Gras | first=Georges | title=Class field theory. From theory to practice | series=Springer Monographs in Mathematics | location=Berlin | publisher=[[Springer-Verlag]] | year=2003 | isbn=3-540-44133-6 | zbl=1019.11032 | pages=204-207 }}
+*{{citation
+  | last1 = Ireland  | first1 = Kenneth
+  | last2 = Rosen  | first2 = Michael
+  | title = A Classical Introduction to Modern Number Theory (Second edition)
+  | publisher = [[Springer Science+Business Media]]
+  | location = New York
+  | year = 1990
+  | isbn = 0-387-97329-X}}
+*{{citation | last1 = Lemmermeyer  | first1 = Franz | title = Reciprocity Laws: from Euler to Eisenstein | publisher = [[Springer Science+Business Media]] | location = Berlin | year = 2000 | isbn = 3-540-66957-4 | doi=10.1007/978-3-662-12893-0 | zbl=0949.11002 | mr=1761696 }}
+* {{citation | last=Neukirch | first=Jürgen | author-link=Jürgen Neukirch | title=Algebraic number theory | others=Translated from the German by Norbert Schappacher | series=Grundlehren der Mathematischen Wissenschaften | volume=322 | location=Berlin | publisher=[[Springer-Verlag]] | year=1999 | isbn=3-540-65399-6 | zbl=0956.11021 }}
+<!--
+Material to be included
+===''n''<sup>th</sup> power residue symbol===
+Let ''k'' be an [[algebraic number field]] with [[ring of integers]]  &nbsp; <math>\mathcal{O}_k,</math> &nbsp;  and let  &nbsp; <math>\mathfrak{p} \subset \mathcal{O}_k </math>  &nbsp; be a [[Number_field#Prime_ideals|prime ideal]]. The [[Norm of an ideal|norm]] of &nbsp;<math>\mathfrak{p} </math>&nbsp; is defined as the cardinality of the residue class ring (since <math>\mathfrak{p} </math>  is prime this is a [[finite field]])  &nbsp; <math> \mathcal{O}_k / \mathfrak{p}\;:\;\;\; \mathrm{N} \mathfrak{p} = |\mathcal{O}_k / \mathfrak{p}|.</math>
+Assume that a primitive ''n''<sup>th</sup> [[root of unity]]  &nbsp; <math>\zeta_n\in\mathcal{O}_k,</math>  &nbsp; and that ''n'' and <math>\mathfrak{p} </math> are [[coprime]] (i.e.&nbsp;<math>n\not\in \mathfrak{p}.</math>) &nbsp; Then
+'''No two distinct ''n''<sup>th</sup> roots of unity can be congruent '''<math>\pmod\mathfrak{p}.</math>
+The proof is by contradiction: assume otherwise, that   &nbsp; <math>\zeta_n^r\equiv\zeta_n^s\pmod\mathfrak{p}, \;\;0 <r<s\le n.</math> &nbsp; Then letting &nbsp; <math>t=s-r,\;\;\zeta_n^t\equiv 1 \pmod\mathfrak{p}, </math> &nbsp; and &nbsp;<math> 0 < t< n.\  </math> From the definition of roots of unity,
+:<math>x^n-1=(x-1)(x-\zeta_n)(x-\zeta_n^2)\dots(x-\zeta_n^{n-1}),</math>  &nbsp; and dividing by  ''x'' &minus; 1  gives
+:<math>x^{n-1}+x^{n-2}+\dots +x + 1 =(x-\zeta_n)(x-\zeta_n^2)\dots(x-\zeta_n^{n-1}).</math>
+Letting ''x'' = 1 and taking residues  <math>\pmod\mathfrak{p},</math>
+:<math>n\equiv(1-\zeta_n)(1-\zeta_n^2)\dots(1-\zeta_n^{n-1})\pmod\mathfrak{p}.</math>
+Since  ''n'' and <math> \mathfrak{p}</math>&nbsp; are coprime,&nbsp;<math> n\not\equiv 0\pmod\mathfrak{p},</math>  &nbsp; but under the assumption, one of the factors on the right must be zero. Therefore the assumption that two distinct roots are congruent is false.
+Thus the residue classes of  &nbsp; <math> \mathcal{O}_k / \mathfrak{p}</math>  &nbsp; containing the powers of &zeta;<sub>''n''</sub> are a subgroup of order ''n'' of its (multiplicative) group of units,  &nbsp; <math>(\mathcal{O}_k/\mathfrak{p}) ^\times  = \mathcal{O}_k /\mathfrak{p}- \{0\}.</math>  &nbsp; Therefore the order of &nbsp;   <math>(\mathcal{O}_k/\mathfrak{p})^ \times</math>  &nbsp; is a multiple of ''n'', and
+:<math>\mathrm{N} \mathfrak{p} = |\mathcal{O}_k / \mathfrak{p}| = |(\mathcal{O}_k / \mathfrak{p} )^\times| + 1 \equiv 1 \pmod{n}.</math>
+There is an analogue of Fermat's theorem in &nbsp; <math>\mathcal{O}_k:</math>&nbsp; If &nbsp;  <math>\alpha \in \mathcal{O}_k,\;\;\; \alpha\not\in \mathfrak{p},</math> &nbsp; then<ref>Lemmermeyer, Ch. 4.1</ref>
+:<math>\alpha^{\mathrm{N} \mathfrak{p} -1}\equiv 1 \pmod{\mathfrak{p} },
+</math> &nbsp;  and since &nbsp;  <math>\mathrm{N} \mathfrak{p} \equiv 1 \pmod{n},</math>
+:<math>\alpha^{\frac{\mathrm{N} \mathfrak{p} -1}{n}}\equiv \zeta_n^s\pmod{\mathfrak{p} }
+</math> &nbsp; is well-defined and congruent to a unique ''n''<sup>th</sup> root of unity &zeta;<sub>''n''</sub><sup>''s''</sup>.
+This root of unity is called  the '''n<sup>th</sup>-power residue symbol for  &nbsp; <math>\mathcal{O}_k,</math>''' &nbsp;  and is denoted by
+:<math>
+\left(\frac{\alpha}{\mathfrak{p} }\right)_n= \zeta_n^s \equiv \alpha^{\frac{\mathrm{N} \mathfrak{p} -1}{n}}\pmod{\mathfrak{p}}.
+</math>
+It can be proven that<ref>Lemmermeyer, Prop 4.1</ref>
+:<math>
+\left(\frac{\alpha}{\mathfrak{p} }\right)_n= 1 \mbox{ if and only if there is an } \eta \in\mathcal{O}_k\;\;\mbox{ such that } \;\;\alpha\equiv\eta^n\pmod{\mathfrak{p}}.</math>
+-->
+[[Category:Algebraic number theory]]

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Latest revision as of 06:06, 13 July 2013

Background and notation

Definition

Properties

Relation to the Hilbert symbol

Generalizations

Power reciprocity law

See also

Notes

References

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